How are the sums of $n$, $2n$ and $3n$ terms connected here?

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1. If \(S_1\), \(S_2\) and \(S_3\) are the sums of \(n\), \(2n\) and \(3n\) terms of an arithmetic progression, show that \(S_3=3(S_2-S_1)\).

Is there a formula for the sum of the first \(n\) terms of an arithmetic progression that we could use?

2. Find the sum of \(n\) terms of the series \[1+\frac{3x}{(1+2x^2)}+\frac{9x^2}{(1+2x^2)^2}+\frac{27x^3}{(1+2x^2)^3}+....\] For what values of \(x\) does this series have a sum to infinity?

What kind of series is this? When does this kind of series have a sum to infinity?